Tunnelling through randomly inhomogeneous barriers as a special problem of the scattering theory

The tunnelling of electrons through an inhomogeneous delta barrier is considered. The strength of the barrier is defined as a function oscillating around a constant mean value along a plane. Owing to deviations from this value, the tunnelling through such a delta barrier has to be interpreted as a scattering. A simple model is discussed when circular windows of a given radius b representing themselves delta barriers of a given strength γ0 are embedded in a homogeneous delta barrier defined with another strength γ¯. When the centers of the windows are distributed randomly in the barrier plane, the potential energy of the electrons is a random function of two space coordinates. The perpendicular incidence is discussed with emphasis on the angular probability density of the tunnelled electrons. The derivation of the angular probability density proves that three basic quantum-mechanical phenomena can be described by one simple formula: tunnelling, diffraction and scattering.